Variations of the total mixed scalar curvature of a distribution

TL;DR

This paper studies the critical points of the total mixed scalar curvature functional of a distribution on a pseudo-Riemannian manifold, deriving Euler-Lagrange equations under various constraints and exploring examples related to different geometric structures.

Contribution

It develops variational formulas and Euler-Lagrange equations for the total mixed scalar curvature of distributions, including volume-preserving and partial metric variations, with applications to diverse geometric contexts.

Findings

Derived Euler-Lagrange equations for critical metrics.

Identified solutions related to contact and 3-Sasakian manifolds.

Found examples of critical metrics with specific geometric properties.

Abstract

We examine the total mixed scalar curvature of a fixed distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution to find the critical points of this action. Together with the arbitrary variations of the metric, we consider also variations that preserve the volume of the manifold or partially preserve the metric (e.g. on the distribution). For each of those cases we obtain the Euler-Lagrange equation and its several solutions. The examples of critical metrics that we find are related to various fields of geometry such as contact and 3-Sasakian manifolds, geodesic Riemannian flows and codimension-one foliations of interesting geometric properties (e.g. totally umbilical, minimal).